The IntegerRelations package contains two routines, LLL, and PSLQ, which are used to solve specific computational problems. LLL is the Lenstra, Lenstra, Lovasz lattice basis reduction. PSLQ is Bailey and Ferguson's partial sum of least squares algorithm. The LinearDependency routine is a user-level routine for applying PSLQ or LLL to solve the integer relation problem, defined as follows.

Given decimal approximations for real or complex numbers , find an integer relation between them, that is, find integers such that is small, if such exist.

The identify Command and the IntegerRelations Package

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The identify command uses the LLL and PSLQ routines to identify exact constants from decimal numbers, for example, given outputs

The process is described as follows. Consider . The identify command first tests if is close to a small rational constant. Next identify tests if is close to an algebraic number. To do this, identify first tests if is a root of a quadratic polynomial. It computes . PSLQ outputs an integer relation with small integer coefficients satisfying , which is small. From this relation you have the minimal polynomial for , namely . The identify command then solves for to obtain from which it determines that . The algorithms in the identify routine can find other relations, for example:

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with(IntegerRelations);

(1)

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Digits := 20;

(2)

>

x := 0.31783724519578224473;

(3)

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PSLQ([1,x,x^2,x^3,x^4]);

(4)

>

solve(y^4-10*y^2+1, y);

(5)

>

identify(x);

(6)

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Each command in the IntegerRelations package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

As the underlying implementation of the IntegerRelations package is a module, it is also possible to use the form IntegerRelations:-command to access a command from the package. For more information, see Module Members.